Topology-preserving digitization of n-dimensional objects by constructing cubical models

TL;DR

This paper introduces a novel cubical space model for digitizing n-dimensional objects, ensuring topological properties are preserved during the conversion from continuous to digital representations.

Contribution

It defines simple n-cubes and proves their topological invariance, enabling topology-preserving digitization of n-dimensional objects using cubical models.

Findings

Homotopy type remains unchanged when deleting or attaching simple n-cubes.

A procedure for constructing compressed, topology-preserving cubical and digital models.

The model effectively represents continuous objects in discrete digital form.

Abstract

This paper proposes a new cubical space model for the representation of continuous objects and surfaces in the n-dimensional Euclidean space by discrete sets of points. The cubical space model concerns the process of converting a continuous object in its digital counterpart, which is a graph, enabling us to apply notions and operations used in digital imaging to cubical spaces. We formulate a definition of a simple n-cube and prove that deleting or attaching a simple n-cube does not change the homotopy type of a cubical space. Relying on these results, we design a procedure, which preserves basic topological properties of an n-dimensional object, for constructing compressed cubical and digital models.